Clifford algebra
Clifford algebra is a type of algebra in which the the geometric product is defined. Quick overview The geometric product is one way of generalizing the concept of complex numbers into higher dimensions. The geometric product is a multivector. In three dimensions a multivector is any sum of a scalar, vector, bivector (a pseudovector), and a trivector (a pseudoscalar). In three dimensions there exists a certain unit trivector (e1∧e2∧e3 = e123 = ) whose geometric product with itself is -1. (Multiplying by the equivalent of i converts anything, including itself, to its dual)Some books say to divide by but this only has the effect of changing the sign since 1/ = / 2 = /-1 = - . Therefore in three dimensions this unit trivector is the Clifford algebra equivalent of i. In two dimensions a certain unit bivector would be the equivalent of i. A unit bivector represents a 90-degree turn so the square of a unit bivector would be a 180-degree turn. Geometric product of vectors A and B = AB = A•B + A∧B :A•B is the dot product of A and B which is a scalar. :A∧B is the wedge product of A and B which is a bivector. Therefore in two Dimensions the geometric product of 2 vectors is a scalar plus a bivector and is therefore the clifford algebra equivalent of a complex number. In two dimensions AB = A•B + A∧B = A•B + (A × B) = ||a|| ||b|| ( cos(θ) + sin(θ) ) = re θ :(A × B) is the 2-D cross product of A and B which is a scalar. Formulas in 3 dimensions Clifford algebra in 3 dimensions is called Cl3. e1, e2, and e3 are the basis vectors. A, B, and C, are unit vectors :A = a1e1 + a2e2 + a3e3 :B = b1e1 + b2e2 + b3e3 :C = c1e1 + c2e2 + c3e3 Wedge product: :3∧5 = 15 :A∧A = 0 :A∧B = bivector (a 2-blade) ::A∧B = (a2b3-a3b2)e23 + (a3b1-a1b3)e31 + (a1b2-a2b1)e12 :A∧B∧C = trivector (a 3-blade) ::A∧B∧C = (a1b2c3 + a2b3c1 + a3b1c2 - a1b3c2 - a2b1c3 - a3b2c1)(e1∧e2∧e3) :(A∧B∧C) • C = A∧B Since A∧A = 0 :0 = (A+B)∧(A+B) :0 = A∧A + A∧B + B∧A + B∧B :0 = 0 + A∧B + B∧A + 0 :0 = A∧B + B∧A Therefore: :-(B∧A) = A∧B This means that rotation from B to A is the negative of rotation from A to B. For Bivectors A and B: :Commutator product = A×B = -(B×A) = ½(AB - BA) :AB = A•B + A×B + A∧B Formulas involving antivectors :See also: Grassmann algebra ē1, ē2, and ē3 are psuedo-vectors or anti-vectors: :ē1 = e2∧e3 :ē2 = e3∧e1 :ē3 = e1∧e2 (a1e1 + a2e2 + a3e3)∧(b1ē1 + b2ē2 + b3ē3) = (a1b1 + a2b2 + a3b3)(e1∧e2∧e3) = (A•B) A•B = a1b1 + a2b2 + a3b3 AA = A•A The Antiwedge product "∨" operates on antivectors: :ē1∨ē2 = (e2∧e3)∨(e3∧e1) = e3 :ē2∨ē3 = (e3∧e1)∨(e1∧e2) = e1 :ē3∨ē1 = (e1∧e2)∨(e2∧e3) = e2 Use in physics When the electromagnetic field is defined as the multivector sum of an electric field vector and a magnetic field bivector, the four Maxwell equations can be reduced to a single equation. Notes See also *Geometric algebra References *Electromagnetism using Geometric Algebra versus Components *http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf *http://wiki.c2.com/?CliffordAlgebra *clifford-algebra-a-visual-introduction (uses an asterick to the geometric product)